Gravity as a Double Copy of Gauge Theory

Transcription

1 Gravity as a Double Copy of Gauge Theory Ricardo Monteiro Queen Mary University of London Iberian Strings 18 January 2017, IST Lisbon Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 1 / 28

2 Motivation 1. Feynman rules for GR: expand Einstein-Hilbert Lagrangian [DeWitt 66] + infinite number of higher-point vertices GR and YM are fundamental theories. Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 2 / 28

3 Gravity YM 2 Free fields Polarisation states: ε µν = ɛ µ ɛ ν (graviton + dilaton + B-field) Degrees of freedom match: (D 2) 2. Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 3 / 28

4 Gravity YM 2 Free fields Polarisation states: ε µν = ɛ µ ɛ ν (graviton + dilaton + B-field) Degrees of freedom match: (D 2) 2. Scattering amplitudes Factorisation of ɛ µ, ɛ ν preserved by interactions! Double copy A grav (ε µν i ) (prop) 1 A YM (ɛ µ i ) A YM ( ɛ ν i ) colour stripped Well established at tree level. Useful but unproven at loop level. Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 3 / 28

5 Gravity YM 2 Free fields Polarisation states: ε µν = ɛ µ ɛ ν (graviton + dilaton + B-field) Degrees of freedom match: (D 2) 2. Scattering amplitudes Factorisation of ɛ µ, ɛ ν preserved by interactions! Double copy A grav (ε µν i ) (prop) 1 A YM (ɛ µ i ) A YM ( ɛ ν i ) colour stripped Well established at tree level. Useful but unproven at loop level. Classical solutions Suggests correspondence between classical theories. Map between solutions? Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 3 / 28

6 Outline Double copy for scattering amplitudes KLT relations Colour-kinematics duality Scattering equations Double copy for classical solutions Exact double copy: Kerr-Schild spacetimes Perturbative double copy Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 4 / 28

7 Double copy for amplitudes Double copy for scattering amplitudes Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 5 / 28

8 Double copy for amplitudes I. KLT relations: string theory origin [Kawai, Lewellen, Tye 86] Vertex operators: V closed (ε µν = ɛ µ ɛ ν ) V open (ɛ µ ) V open ( ɛ ν ) s ij = (k i + k j ) 2 A grav 3 = A YM (123) ÃYM (123) A grav 4 = sin πα s 12 πα A YM (1234) ÃYM (1243) Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 6 / 28

9 Double copy for amplitudes I. KLT relations: string theory origin [Kawai, Lewellen, Tye 86] Vertex operators: V closed (ε µν = ɛ µ ɛ ν ) V open (ɛ µ ) V open ( ɛ ν ) s ij = (k i + k j ) 2 A grav 3 = A YM (123) ÃYM (123) A grav 4 = sin πα s 12 πα A YM (1234) ÃYM (1243) Field theory limit is α 0. In general (tree level) [Bern, Dixon, Perelstein, Rozowsky 98] An grav = A YM (P n ) S KLT [P n, P n] Ã YM (P n) P n,p n Useful at loop level via unitarity cuts. S KLT s n 3 ij Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 6 / 28

10 Double copy for amplitudes I. KLT relations: string theory origin [Kawai, Lewellen, Tye 86] Vertex operators: V closed (ε µν = ɛ µ ɛ ν ) V open (ɛ µ ) V open ( ɛ ν ) s ij = (k i + k j ) 2 A grav 3 = A YM (123) ÃYM (123) A grav 4 = sin πα s 12 πα A YM (1234) ÃYM (1243) Field theory limit is α 0. In general (tree level) [Bern, Dixon, Perelstein, Rozowsky 98] An grav = A YM (P n ) S KLT [P n, P n] Ã YM (P n) P n,p n Useful at loop level via unitarity cuts. S KLT s n 3 ij Recall YM colour decomposition: colour traces or colour factors. An YM = A YM (1, 2,..., n) tr(t a 1 T a2 T an ) = N α c α non cyclic with c α = f abc f f, f abc = tr([t a, T b ]T c ), but Jacobi identities: c α ± c β ± c γ = 0 α cubic Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 6 / 28

11 Double copy for amplitudes II. BCJ duality and double copy [Bern, Carrasco, Johansson 08, 10] Gauge theory A YM = n α c α D α α cubic kinematic numerators: n α (k i, ɛ i ) colour factors: c α = f abc f f propagators: D α = r K α,r 2 Colour-kinematics duality n α : c α ± c β ± c γ = 0 n α ± n β ± n γ = 0 Gravity A grav = α cubic n α ñ α D α Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 7 / 28

12 Double copy for amplitudes II. BCJ duality and double copy [Bern, Carrasco, Johansson 08, 10] Gauge theory A YM = n α c α D α Gravity A grav = α cubic α cubic n α ñ α D α kinematic numerators: n α (k i, ɛ i ) colour factors: c α = f abc f f propagators: D α = r K 2 α,r Colour-kinematics duality n α : c α ± c β ± c γ = 0 n α ± n β ± n γ = 0 Double copy to gravity colour-kinematics satisfying n α same propagators states scattered: ε µν = ɛ µ ɛ ν Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 7 / 28

13 Double copy for amplitudes II. BCJ duality and double copy [Bern, Carrasco, Johansson 08, 10] Gauge theory A YM = n α c α D α Gravity A grav = α cubic α cubic n α ñ α D α kinematic numerators: n α (k i, ɛ i ) colour factors: c α = f abc f f propagators: D α = r K 2 α,r Colour-kinematics duality n α : c α ± c β ± c γ = 0 n α ± n β ± n γ = 0 Double copy to gravity colour-kinematics satisfying n α same propagators states scattered: ε µν = ɛ µ ɛ ν Stringy understanding from monodromy [Bjerrum-Bohr, Damgaard, Vanhove; Stieberger 09] and pure spinor formalism. [Mafra, Schlotterer, Stieberger 11] Self-dual YM and gravity: kinematic algebra, off-shell BCJ. [RM, O Connell 11] Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 7 / 28

14 Double copy for amplitudes II. BCJ duality and double copy [Bern, Carrasco, Johansson 08, 10] Gauge theory A YM = n α c α D α Gravity A grav = α cubic α cubic n α ñ α D α kinematic numerators: n α (k i, ɛ i ) colour factors: c α = f abc f f propagators: D α = r K 2 α,r Colour-kinematics duality n α : c α ± c β ± c γ = 0 n α ± n β ± n γ = 0 Double copy to gravity colour-kinematics satisfying n α same propagators states scattered: ε µν = ɛ µ ɛ ν Tree level: duality true, double copy same as KLT relations. Loop level (integrand): duality conjectural, double copy gets unitarity cuts. Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 8 / 28

15 Double copy for amplitudes Gravity YM ỸM : multiplication table (N = 4 SYM) (N = 4 SYM) (N = 8 SUGRA) (N = 4 SYM) (N = 2 SYM) (N = 6 SUGRA) (N = 4 SYM) (N = 1 SYM) (N = 5 SUGRA) (N = 4 SYM) (N = 0 YM) (N = 4 SUGRA) (N = 2 SYM) (N = 2 SYM) (N = 4 SUGRA) + 2 vect.multipl. etc. (N = 0 YM) (N = 0 YM) (N = 0 SUGRA) Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 9 / 28

16 Double copy for amplitudes Gravity YM ỸM : multiplication table (N = 4 SYM) (N = 4 SYM) (N = 8 SUGRA) (N = 4 SYM) (N = 2 SYM) (N = 6 SUGRA) (N = 4 SYM) (N = 1 SYM) (N = 5 SUGRA) (N = 4 SYM) (N = 0 YM) (N = 4 SUGRA) (N = 2 SYM) (N = 2 SYM) (N = 4 SUGRA) + 2 vect.multipl. etc. (N = 0 YM) (N = 0 YM) (N = 0 SUGRA) S N =0 SUGRA = d D x [ 2 g κ 2 R 1 2(D 2) µ ϕ µϕ 1 ] 2κϕ 6 e D 2 H λµν H λµν, H = db. Pure Einstein gravity? tree level: yes if external particles are all gravitons loop level: need to project out dilaton ϕ and B-field in loops Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 9 / 28

17 Double copy for amplitudes Gravity YM ỸM : multiplication table (N = 4 SYM) (N = 4 SYM) (N = 8 SUGRA) (N = 4 SYM) (N = 2 SYM) (N = 6 SUGRA) (N = 4 SYM) (N = 1 SYM) (N = 5 SUGRA) (N = 4 SYM) (N = 0 YM) (N = 4 SUGRA) (N = 2 SYM) (N = 2 SYM) (N = 4 SUGRA) + 2 vect.multipl. etc. (N = 0 YM) (N = 0 YM) (N = 0 SUGRA) S N =0 SUGRA = d D x [ 2 g κ 2 R 1 2(D 2) µ ϕ µϕ 1 ] 2κϕ 6 e D 2 H λµν H λµν, H = db. Pure Einstein gravity? tree level: yes if external particles are all gravitons loop level: need to project out dilaton ϕ and B-field in loops UV surprises in 4D SUGRA: finite for N > 4? [Bern, Carrasco, Dixon, Johansson, Roiban ] N = 8 SUGRA as same D crit as N = 4 SYM up to 4 loops. Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 9 / 28

18 Double copy for amplitudes III. Scattering equations and CHY formulas [Cachazo, He, Yuan 13] Consider n massless particles, k 2 i = 0, i = 1,..., n E i = j i k i k j σ i σ j = 0, i kinematic invariants s ij = 2 k i k j points σ i CP 1 i k i = 0 : SL(2, C) invariance, σ A σ + B C σ + D (n 3)! solutions σ (A) i factorisation: (k k m ) 2 0 gives σ 1,..., σ m σ 1 m Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 10 / 28

19 Double copy for amplitudes III. Scattering equations and CHY formulas [Cachazo, He, Yuan 13] Consider n massless particles, k 2 i = 0, i = 1,..., n E i = j i k i k j σ i σ j = 0, i kinematic invariants s ij = 2 k i k j points σ i CP 1 i k i = 0 : SL(2, C) invariance, σ A σ + B C σ + D (n 3)! solutions σ (A) i factorisation: (k k m ) 2 0 gives σ 1,..., σ m σ 1 m Also apear in high-energy fixed-angle string scattering. [Gross, Mende 88] Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 10 / 28

20 Double copy for amplitudes III. Scattering equations and CHY formulas [Cachazo, He, Yuan 13] Tree-level scattering amplitude: A = dµ I Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 11 / 28

21 Double copy for amplitudes III. Scattering equations and CHY formulas [Cachazo, He, Yuan 13] Tree-level scattering amplitude: A = dµ I measure is universal d n σ dµ = δ(e i ) = A = volsl(2, C) i (n 3)! A=1 I J σ=σ (A) Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 11 / 28

22 Double copy for amplitudes III. Scattering equations and CHY formulas [Cachazo, He, Yuan 13] Tree-level scattering amplitude: A = dµ I measure is universal d n σ dµ = δ(e i ) = A = volsl(2, C) I specifies the theory ( tr(t I YM = Pf a 1 T a2 T an ) M(ɛ) σ 12 σ 23 σ n1 i ) + non-cyclic perm I grav = Pf M(ɛ) Pf M( ɛ) Gravity YM 2 (n 3)! A=1 I J σ=σ (A) σ rs = σ r σ s Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 11 / 28

23 Double copy for amplitudes III. Scattering equations and CHY formulas [Cachazo, He, Yuan 13] Tree-level scattering amplitude: A = dµ I measure is universal d n σ dµ = δ(e i ) = A = volsl(2, C) I specifies the theory ( tr(t I YM = Pf a 1 T a2 T an ) M(ɛ) σ 12 σ 23 σ n1 i ) + non-cyclic perm I grav = Pf M(ɛ) Pf M( ɛ) Gravity YM 2 (n 3)! A=1 I J σ=σ (A) σ rs = σ r σ s Predecessor: 4D formula from twistor-string theory. [Witten 03; Roiban, Spradlin, Volovich 04] Worldsheet model for CHY: ambitwistor-string theory. [Mason, Skinner 13] Loop-level formulas! [Geyer, Mason, RM, Tourkine 15, 16] $ [Adamo, Casali, Skinner 13] Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 11 / 28

24 Double copy for amplitudes Summary for amplitudes A grav (ε µν i ) (prop) 1 A YM (ɛ µ i ) A YM ( ɛ ν i ) colour stripped KLT relations BCJ double copy A YM = A grav = Pn,P n α cubic A YM (ɛ, P n ) S KLT [P n, P n] A YM ( ɛ, P n) n α (ɛ) c α D α A grav = α cubic n α (ɛ) n α ( ɛ) D α CHY formulas A = dµ I I YM = Pf M(ɛ) C I grav = Pf M(ɛ) Pf M( ɛ) Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 12 / 28

25 Classical double copy Double copy for classical solutions Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 13 / 28

26 Classical double copy Double copy for black holes? [RM, O Connell, White 14] Question: is there a double copy for classical solutions? Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 14 / 28

27 Classical double copy Double copy for black holes? [RM, O Connell, White 14] Question: is there a double copy for classical solutions? Challenges: What is graviton in exact solution? Non-perturbative double copy? Relation of diffeo. choice in gravity to gauge choice in YM? Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 14 / 28

28 Classical double copy Double copy for black holes? [RM, O Connell, White 14] Question: is there a double copy for classical solutions? Challenges: What is graviton in exact solution? Non-perturbative double copy? Relation of diffeo. choice in gravity to gauge choice in YM? Still... Should work in perturbation theory. Some solutions can be constructed perturbatively. Examples: Schwarzchild [Duff 73; Neill, Rothstein 13], shockwave [Saotome, Akhoury 12]. Direct map of exact solutions? Need miracle! Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 14 / 28

29 Classical double copy Double copy for black holes? [RM, O Connell, White 14] Question: is there a double copy for classical solutions? Challenges: What is graviton in exact solution? Non-perturbative double copy? Relation of diffeo. choice in gravity to gauge choice in YM? Still... Should work in perturbation theory. Some solutions can be constructed perturbatively. Examples: Schwarzchild [Duff 73; Neill, Rothstein 13], shockwave [Saotome, Akhoury 12]. Direct map of exact solutions? Need miracle! Symmetry Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 14 / 28

30 Classical double copy: exact Stationary Kerr-Schild spacetimes Exact perturbation g µν = η µν + φ k µ k ν where k µ is null and geodesic wrt η µν and g µν. (k µ = g µν k ν = η µν k ν ) Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 15 / 28

31 Classical double copy: exact Stationary Kerr-Schild spacetimes Exact perturbation g µν = η µν + φ k µ k ν where k µ is null and geodesic wrt η µν and g µν. (k µ = g µν k ν = η µν k ν ) Einstein equations linearise: g µν = η µν φ k µ k ν R µ ν = 1 2 α [ µ (φk α k ν ) + ν (φk α k µ ) α (φk µ k ν )] µ η µν ν Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 15 / 28

32 Classical double copy: exact Stationary Kerr-Schild spacetimes Exact perturbation g µν = η µν + φ k µ k ν where k µ is null and geodesic wrt η µν and g µν. (k µ = g µν k ν = η µν k ν ) Einstein equations linearise: g µν = η µν φ k µ k ν R µ ν = 1 2 α [ µ (φk α k ν ) + ν (φk α k µ ) α (φk µ k ν )] µ η µν ν Stationary vacuum case (take k 0 = 1): R 0 0 = φ = 0 R i 0 = 1 2 [ ( l i φk l) ( l φk i)] = 0 Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 15 / 28

33 Classical double copy: exact Stationary Kerr-Schild spacetimes Exact perturbation g µν = η µν + φ k µ k ν where k µ is null and geodesic wrt η µν and g µν. (k µ = g µν k ν = η µν k ν ) Einstein equations linearise: g µν = η µν φ k µ k ν R µ ν = 1 2 α [ µ (φk α k ν ) + ν (φk α k µ ) α (φk µ k ν )] µ η µν ν Stationary vacuum case (take k 0 = 1): R 0 0 = φ = 0 R i 0 = 1 2 [ ( l i φk l) ( l φk i)] = 0 BCJ single copy linear Abelian, A a µ = φ k µ c a c a const Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 15 / 28

34 Classical double copy: exact Stationary Kerr-Schild spacetimes Exact perturbation g µν = η µν + φ k µ k ν where k µ is null and geodesic wrt η µν and g µν. (k µ = g µν k ν = η µν k ν ) Einstein equations linearise: g µν = η µν φ k µ k ν R µ ν = 1 2 α [ µ (φk α k ν ) + ν (φk α k µ ) α (φk µ k ν )] µ η µν ν Stationary vacuum case (take k 0 = 1): R 0 0 = φ = 0 R i 0 = 1 2 [ ( l i φk l) ( l φk i)] = 0 BCJ single copy linear Abelian, A a µ = φ k µ c a c a const { 0 = D µ F a µν = c a 2 [ φ ν = 0 ( l i φk l) ( l φk i)] ν = i Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 15 / 28

35 Classical double copy: exact Simplest example: point charge Check spherically symmetric solutions sourced by point charge. Einstein theory: Schwarzschild solution g µν = η µν + φ k µ k ν, φ(r) = 2M r, k = dt + dr Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 16 / 28

36 Classical double copy: exact Simplest example: point charge Check spherically symmetric solutions sourced by point charge. Einstein theory: Schwarzschild solution g µν = η µν + φ k µ k ν, φ(r) = 2M r, k = dt + dr YM theory: Coulomb solution A a µ = φ k µ c a, φ(r) = q r A a A a + d( q c a log r) = q r ca dt Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 16 / 28

37 Classical double copy: exact Simplest example: point charge Check spherically symmetric solutions sourced by point charge. Einstein theory: Schwarzschild solution g µν = η µν + φ k µ k ν, φ(r) = 2M r, k = dt + dr YM theory: Coulomb solution A a µ = φ k µ c a, φ(r) = q r A a A a + d( q c a log r) = q r ca dt Schwarzschild (Coulomb) 2 Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 16 / 28

38 Classical double copy: exact Simplest example: point charge Check spherically symmetric solutions sourced by point charge. Einstein theory: Schwarzschild solution g µν = η µν + φ k µ k ν, φ(r) = 2M r, k = dt + dr YM theory: Coulomb solution A a µ = φ k µ c a, φ(r) = q r A a A a + d( q c a log r) = q r ca dt Schwarzschild (Coulomb) 2 Makes sense! However, expect (YM) 2 Einstein g µν + dilaton ϕ + B-field B µν Why vacuum? How to add dilaton? See later. Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 16 / 28

39 Classical double copy: exact Stationary Kerr-Schild: more examples Kerr solution: (M, a), a = J/M. Schwarzschild is a = 0. φ(r, z) = 2 M r 3 r 4 + a 2 z 2, x 2 + y 2 r 2 + a 2 + z2 r 2 = 1 k = dt + rx + ay ry ax r 2 dx + + a2 r 2 + a 2 dy + z r dz Single copy: Maxwell field generated by certain rotating charged disk. Extends to D > 4: Myers-Perry black holes (M, a i ) are also Kerr-Schild. But there are other black holes families! Black rings, etc... Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 17 / 28

40 Classical double copy: exact Stationary Kerr-Schild: more examples Kerr solution: (M, a), a = J/M. Schwarzschild is a = 0. φ(r, z) = 2 M r 3 r 4 + a 2 z 2, x 2 + y 2 r 2 + a 2 + z2 r 2 = 1 k = dt + rx + ay ry ax r 2 dx + + a2 r 2 + a 2 dy + z r dz Single copy: Maxwell field generated by certain rotating charged disk. Extends to D > 4: Myers-Perry black holes (M, a i ) are also Kerr-Schild. But there are other black holes families! Black rings, etc... Cosmological constant constant charge density. [Luna, RM, O Connell, White 15] NUT charge magnetic monopole: multi-kerr-schild g µν = η µν + φ k µ k ν + ψ l µ l ν, φ M, ψ N A dyon µ = φ k µ + ψ l µ Radiation from an accelerated particle: correct Bremsstrahlung. [Luna, RM, Nicholson, O Connell, White 16] Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 17 / 28

41 Spinorial double copy Alternative formulation [in preparation, Luna et al] Try double copy of curvatures: A µ = ɛ µ e ik x, F µν = i(k µ ɛ ν k ν ɛ µ ) e ik x h µν = ɛ µ ɛ ν e ik x, R µνρλ = 1 2 (k µɛ ν k ν ɛ µ )(k ρ ɛ λ k λ ɛ ρ ) e ik x Obvious relation: e ik x R µνρλ F µν F ρλ More general? Not so simple: symmetries of R µνρλ, non-linear gauge,... Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 18 / 28

42 Spinorial double copy Alternative formulation [in preparation, Luna et al] Try double copy of curvatures: A µ = ɛ µ e ik x, F µν = i(k µ ɛ ν k ν ɛ µ ) e ik x h µν = ɛ µ ɛ ν e ik x, R µνρλ = 1 2 (k µɛ ν k ν ɛ µ )(k ρ ɛ λ k λ ɛ ρ ) e ik x Obvious relation: e ik x R µνρλ F µν F ρλ More general? Not so simple: symmetries of R µνρλ, non-linear gauge,... Spinorial approach to GR (D = 4) [Penrose 60] ( ) Basic object is σ µ such that σ µ + AȦ AȦσν BḂ σν AȦσµ εȧḃ = g µν ε BḂ AB Translation spacetime indices spinor indices: V µ V A Ȧ = σµ AȦ V µ. Like spinor-helicity, but curved. Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 18 / 28

43 Spinorial double copy Alternative formulation [in preparation, Luna et al] Try double copy of curvatures: A µ = ɛ µ e ik x, F µν = i(k µ ɛ ν k ν ɛ µ ) e ik x h µν = ɛ µ ɛ ν e ik x, R µνρλ = 1 2 (k µɛ ν k ν ɛ µ )(k ρ ɛ λ k λ ɛ ρ ) e ik x Obvious relation: e ik x R µνρλ F µν F ρλ More general? Not so simple: symmetries of R µνρλ, non-linear gauge,... Spinorial approach to GR (D = 4) [Penrose 60] ( ) Basic object is σ µ such that σ µ + AȦ AȦσν BḂ σν AȦσµ εȧḃ = g µν ε BḂ AB Translation spacetime indices spinor indices: V µ V A Ȧ = σµ AȦ V µ. Like spinor-helicity, but curved. Want formula: curvature R 1 (curvature F)2 scalar Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 18 / 28

44 Spinorial double copy Weyl spinor and algebraic classification Weyl curvature W µνρλ : W µνρλ = R µνρλ + terms(r µν, g µν ) = R µνρλ in vacuum as R µν = 0 Weyl spinor C ABCD : W µνρλ W A ȦBḂCĊDḊ = C ABCD εȧḃ ε ĊḊ + CȦḂĊḊ ε AB ε CD where C ABCD = C (ABCD) and CȦḂĊḊ is complex conjugate. Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 19 / 28

45 Spinorial double copy Weyl spinor and algebraic classification Weyl curvature W µνρλ : W µνρλ = R µνρλ + terms(r µν, g µν ) = R µνρλ in vacuum as R µν = 0 Weyl spinor C ABCD : W µνρλ W A ȦBḂCĊDḊ = C ABCD εȧḃ ε ĊḊ + CȦḂĊḊ ε AB ε CD where C ABCD = C (ABCD) and CȦḂĊḊ is complex conjugate. Can decompose into four rank 1 spinors: C ABCD = a (A b B c C d D) Four principal null directions: a A Ȧ = a A āȧ and same for baȧ, caȧ, daȧ. Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 19 / 28

46 Spinorial double copy Weyl spinor and algebraic classification Weyl curvature W µνρλ : W µνρλ = R µνρλ + terms(r µν, g µν ) = R µνρλ in vacuum as R µν = 0 Weyl spinor C ABCD : W µνρλ W A ȦBḂCĊDḊ = C ABCD εȧḃ ε ĊḊ + CȦḂĊḊ ε AB ε CD where C ABCD = C (ABCD) and CȦḂĊḊ is complex conjugate. Can decompose into four rank 1 spinors: C ABCD = a (A b B c C d D) Four principal null directions: a A Ȧ = a A āȧ and same for baȧ, caȧ, daȧ. Algebraic classification of spacetimes [Petrov 54] How many principal null directions are aligned? Types I, II, D, III, N, O. Type D: a A c A, b A d A, then C ABCD y AB y CD, where y AB = a (A b B). Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 19 / 28

47 Spinorial double copy Spinorial double copy Take Minkowski space: σ a = 1 2 (1, σ i ). Maxwell spinor f AB : F µν F A ȦBḂ = f AB εȧḃ + fȧḃ ε CD where f AB = f (AB) and fȧḃ is complex conjugate. Also f AB = r (A s B). Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 20 / 28

48 Spinorial double copy Spinorial double copy Take Minkowski space: σ a = 1 2 (1, σ i ). Maxwell spinor f AB : F µν F A ȦBḂ = f AB εȧḃ + fȧḃ ε CD where f AB = f (AB) and fȧḃ is complex conjugate. Also f AB = r (A s B). Spinorial double copy C ABCD = 1 S f AB f CD Applies to all previous Kerr-Schild examples (type D). Gives meaning to C ABCD y AB y CD. φ = S + S, where φ is same as Kerr-Schild case. Eom: a a φ = 0. Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 20 / 28

49 Spinorial double copy Spinorial double copy Take Minkowski space: σ a = 1 2 (1, σ i ). Maxwell spinor f AB : F µν F A ȦBḂ = f AB εȧḃ + fȧḃ ε CD where f AB = f (AB) and fȧḃ is complex conjugate. Also f AB = r (A s B). Spinorial double copy C ABCD = 1 S f AB f CD Applies to all previous Kerr-Schild examples (type D). Gives meaning to C ABCD y AB y CD. φ = S + S, where φ is same as Kerr-Schild case. Eom: a a φ = 0. More examples? C-metric Lienard-Weichert potential for uniform acceleration Extension to D > 4? Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 20 / 28

50 Classical double copy: perturbative Perturbative double copy [Luna, RM, Nicholson, Ochirov, O Connell, Westerberger, White 16] In general, no exact solutions, no hope for exact double copy. Need to use perturbation theory. Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 21 / 28

51 Classical double copy: perturbative Perturbative double copy [Luna, RM, Nicholson, Ochirov, O Connell, Westerberger, White 16] In general, no exact solutions, no hope for exact double copy. Need to use perturbation theory. First map linearised solutions. talk by Silvia Nagy [Anastasiou et al, Cardoso et al] Correct order-by-order in BCJ-ish perturbation theory. Translate back to standard gauge only if needed. Recent work: radiation from point charges [Goldberger, Ridgway 16], BPS black holes [Cardoso, Nagy, Nampuri 16]. Prior work: Schwarszchild [Neill, Rothstein 13], shockwave [Saotome, Akhoury 12]. Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 21 / 28

52 Classical double copy: perturbative Perturbative double copy [Luna, RM, Nicholson, Ochirov, O Connell, Westerberger, White 16] In general, no exact solutions, no hope for exact double copy. Need to use perturbation theory. First map linearised solutions. talk by Silvia Nagy [Anastasiou et al, Cardoso et al] Correct order-by-order in BCJ-ish perturbation theory. Translate back to standard gauge only if needed. Recent work: radiation from point charges [Goldberger, Ridgway 16], BPS black holes [Cardoso, Nagy, Nampuri 16]. Prior work: Schwarszchild [Neill, Rothstein 13], shockwave [Saotome, Akhoury 12]. Goals Rewrite gravitational perturbation theory in BCJ-ish way. [Bern, Grant 99] [Cheung, Remmen 16] Clarify splitting into graviton, dilaton, B-field. Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 21 / 28

53 Classical double copy: perturbative Perturbative double copy: setup Construct solutions perturbatively: start with linearised solution j, j = 0. proceed order by order: f (0) = j, f (1) g j 2, f (2) g 2 j 3,... j f (x) = j + j j j + + O(j 4 ) = j f (n) (x) n=0 Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 22 / 28

54 Classical double copy: perturbative Perturbative double copy: setup Construct solutions perturbatively: start with linearised solution j, j = 0. proceed order by order: f (0) = j, f (1) g j 2, f (2) g 2 j 3,... j f (x) = j + j j j + + O(j 4 ) = j f (n) (x) n=0 Gauge theory field A µ, admits BCJ Lagrangian enforcing colour-kinematics duality. 0 Gravity want field H µν graviton + dilaton + B-field, fat graviton whose vertices are double copy of YM with BCJ Lagrangian. Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 22 / 28

55 Classical double copy: perturbative Perturbative double copy: setup Example: 3-point vertex Gauge theory f (1) (x) = A (1)aµ ( p 1 ) = i 2p1 2 f abc d D p 2 d D p D 3 δ (p 1 + p 2 + p 3 ) [ f (0) f (0) (p 1 p 2 ) γ η µβ + (p 2 p 3 ) µ η βγ + (p 3 p 1 ) β η γµ] A (0)b β (p 2)A (0)c γ (p 3 ) Gravity H (1)µµ ( p 1 ) = 1 4p1 2 d D p 2 d D p D 3 δ (p 1 + p 2 + p 3 ) [(p 1 p 2 ) γ η µβ + (p 2 p 3 ) µ η βγ + (p 3 p 1 ) β η γµ] [ (p 1 p 2 ) γ η µ β + (p 2 p 3 ) µ η β γ + (p 3 p 1 ) β η γ µ ] H (0) ββ (p 2 )H (0) γγ (p 3 ) Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 23 / 28

56 Classical double copy: perturbative Perturbative double copy: setup Example: 3-point vertex Gauge theory f (1) (x) = A (1)aµ ( p 1 ) = i 2p1 2 f abc d D p 2 d D p D 3 δ (p 1 + p 2 + p 3 ) [ f (0) f (0) (p 1 p 2 ) γ η µβ + (p 2 p 3 ) µ η βγ + (p 3 p 1 ) β η γµ] A (0)b β (p 2)A (0)c γ (p 3 ) Gravity H (1)µµ ( p 1 ) = 1 4p1 2 d D p 2 d D p D 3 δ (p 1 + p 2 + p 3 ) [(p 1 p 2 ) γ η µβ + (p 2 p 3 ) µ η βγ + (p 3 p 1 ) β η γµ] [ (p 1 p 2 ) γ η µ β + (p 2 p 3 ) µ η β γ + (p 3 p 1 ) β η γ µ ] H (0) ββ (p 2 )H (0) γγ (p 3 ) BCJ Lagrangian for YM has infinite sequence of non-local ( 1 ) vertices. [Bern, Dennen, Huang, Kiermaier 10] Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 23 / 28

57 Classical double copy: perturbative Fat graviton: linearised theory Definition H µν (x) = h µν (x) + B µν (x) + P q µν(ϕ h) ϕ is the dilaton. B µν is the B-field. µ B µν = 0. κ h µν = η µν g g µν = κ (h µν 1 2 η µνh) + O(h 2 ). µ h µν = 0. ( ) Projector: Pµν q = 1 D 2 η µν qµ ν+qν µ q, where q µ = const. Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 24 / 28

58 Classical double copy: perturbative Fat graviton: linearised theory Definition H µν (x) = h µν (x) + B µν (x) + P q µν(ϕ h) ϕ is the dilaton. B µν is the B-field. µ B µν = 0. κ h µν = η µν g g µν = κ (h µν 1 2 η µνh) + O(h 2 ). µ h µν = 0. ( ) Projector: Pµν q = 1 D 2 η µν qµ ν+qν µ q, where q µ = const. Properties H µν has (D 2) 2 propagating dof s. µ H µν = 0. EOMs: 2 H µν = 0 2 h µν = 0, 2 ϕ = 0, 2 B µν = 0 Projector origin: D 2 i=1 ɛi µɛ i ν = η µν qµkν+qνkµ q k, where ɛ i q = 0. Invertible: can get skinny fields h µν, ϕ, B µν from H µν. Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 24 / 28

59 Classical double copy: perturbative Fat graviton: first correction Example Simple starting point: H (0) µν = κ 2 M 4πr u µu ν, with u µ = (1, 0, 0, 0). Compute using double copy of YM 3-point vertex: H (1) (x) = ( κ ) 2 M Result is: H µν (1) 2 = 2 4(4πr) 2 k µk ν, with k µ = (0, x i /r). H (0) H (0) Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 25 / 28

60 Classical double copy: perturbative Fat graviton: first correction Example Simple starting point: H (0) µν = κ 2 M 4πr u µu ν, with u µ = (1, 0, 0, 0). Compute using double copy of YM 3-point vertex: H (1) (x) = ( κ ) 2 M Result is: H µν (1) 2 = 2 4(4πr) 2 k µk ν, with k µ = (0, x i /r). Exact spherically symmetric, static solution of Einstein-dilaton gravity known! Translate the result for comparison: H µν (1) = h (1) µν + B µν (1) + Pµν(φ q (1) h (1) ) + T µν (1) where T (1) µν (h (0), φ (0), B (0) ) encodes gauge transf / field redefinition. To be avoided! Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 25 / 28 H (0) H (0)

61 Classical double copy: perturbative Dilatonic point mass: JNW solution Start with linearised solution: graviton M, dilaton Y. Construct order-by-order in perturbation theory. Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 26 / 28

62 Classical double copy: perturbative Dilatonic point mass: JNW solution Start with linearised solution: graviton M, dilaton Y. Construct order-by-order in perturbation theory. Exact solution (M, Y ) was found by Janis, Newman, Winicour 68: ( ds 2 = 1 ρ ) γ ( 0 dt ρ ) γ ( 0 dρ ρ ) 1 γ 0 ρ 2 dω 2 ρ ρ ρ ϕ = Y ( log 1 ρ ) 0 ρ 0 = 2 M ρ 0 ρ 2 + Y 2 M γ = M2 + Y 2 Y = 0: pure Einstein gravity Schwarzschild black hole (Kerr-Schild) Y 0: naked singularity at origin ρ = ρ 0, cf. no-hair theorems Y = M: matches previous example H (0) µν M/r. Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 26 / 28

63 Classical double copy: perturbative Dilatonic point mass: JNW solution Start with linearised solution: graviton M, dilaton Y. Construct order-by-order in perturbation theory. Exact solution (M, Y ) was found by Janis, Newman, Winicour 68: ( ds 2 = 1 ρ ) γ ( 0 dt ρ ) γ ( 0 dρ ρ ) 1 γ 0 ρ 2 dω 2 ρ ρ ρ ϕ = Y ( log 1 ρ ) 0 ρ 0 = 2 M ρ 0 ρ 2 + Y 2 M γ = M2 + Y 2 Y = 0: pure Einstein gravity Schwarzschild black hole (Kerr-Schild) Y 0: naked singularity at origin ρ = ρ 0, cf. no-hair theorems Y = M: matches previous example H (0) µν M/r. General JNW not vacuum and not Kerr-Schild. Exact double copy? Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 26 / 28

64 Beyond vacuum Beyond vacuum solutions Back to question: (YM) 2 Einstein g µν + dilaton ϕ + B-field Spherically symetric, static solution sourced by point charge: Yang-Mills: Coulomb solution. Pure Einstein gravity: Schwarschild solution. Einstein + dilaton: JNW solution. More general double copy of Coulomb. (M u µu ν + (M Y ) 12 ) (ηµν qµlν qνlµ), q l = 1 A µ (0) a = g ca uµ H(0) µν = κ 4πr 2 1 4πr Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 27 / 28

65 Beyond vacuum Beyond vacuum solutions Back to question: (YM) 2 Einstein g µν + dilaton ϕ + B-field Spherically symetric, static solution sourced by point charge: Yang-Mills: Coulomb solution. Pure Einstein gravity: Schwarschild solution. Einstein + dilaton: JNW solution. More general double copy of Coulomb. (M u µu ν + (M Y ) 12 ) (ηµν qµlν qνlµ), q l = 1 A µ (0) a = g ca uµ H(0) µν = κ 4πr 2 1 4πr Analogue for scattering amplitudes: not all gravity states are of the form ε µν = ɛ µ ɛ ν, but double copy gives full gravity theory, ε µν is linear comb. of above. Eg. dilaton case ε µν ϕ D 2 i=1 ɛi µɛ i ν = η µν qµkν+qνkµ q k Of course, pure Einstein gravity is the most interesting sector. Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 27 / 28

66 Conclusion Conclusion Gravity is double copy from gauge theory, at least perturbatively. Major tool in scattering amplitudes. Double copy of classical solutions is possible. Exact examples: simplest is Schwarzschild (Coulomb) 2. Perturbative double copy based on fat graviton. Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 28 / 28

67 Conclusion Conclusion Gravity is double copy from gauge theory, at least perturbatively. Major tool in scattering amplitudes. Double copy of classical solutions is possible. Exact examples: simplest is Schwarzschild (Coulomb) 2. Perturbative double copy based on fat graviton. Many open questions! Amplitudes: loop-level structure? Exact solutions: Non Kerr-Schild? Non vacuum? Spinorial for D > 4? New GR solutions from gauge theory? Perturbative story: phenomenological applications? Ricardo Monteiro (Queen Mary) Gravity as a Double Copy of Gauge Theory 28 / 28